On the compactness of bi-parameter singular integrals

TL;DR

This work addresses the compactness of bi-parameter Calderón-Zygmund operators on $L^2(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$ and endpoint spaces. It introduces a new bi-parameter $T1$-type compactness theorem under the product weak compactness property, the mixed weak compactness/CMO property, and $T1, T^t1, T_t1, T_t^t1 \in CMO(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$, and proves endpoint compactness from $H^1$ to $L^1$ and from $L^\infty$ to $CMO$. The sufficiency uses a decomposition of $T$ into top-level paraproducts, partial paraproducts, and a fully cancellative bi-parameter CZO, together with a new abstract theory of partially localized operators on tensor products of Hilbert spaces; necessity is established as well. The results sharpen previous conditions, avoid compact full kernel representations, and extend to weighted and endpoint settings, providing a sharp, modular framework for bi-parameter compactness with potential PDE applications.

Abstract

We establish a new $T1$ theorem for the compactness of bi-parameter Calderón-Zygmund singular integral operators. Namely, we show that if a bi-parameter CZO $T$ satisfies the product weak compactness property, the mixed weak compactness/CMO property, and $T1, T^t1,$ $T_t1, T_t^t1 \in \text{CMO}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$, then $T$ is compact on $L^2(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$. We also obtain endpoint compactness results for these operators and use them to deduce the necessity of most of our hypotheses. In particular, our conditions characterize the simultaneous $L^2(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$-compactness of a bi-parameter CZO and its partial transpose. Our assumptions improve upon previously known sufficient conditions, and our proof, which is shorter and simpler than earlier arguments, utilizes a new abstract compactness criterion for partially localized operators on tensor products of Hilbert spaces.

Theorems & Definitions (59)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Definition 2.2
• Proposition 2.3
• proof